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Linear-Degree Extractors and the Inapproximability of Max Clique and Chromatic Number David Zuckerman University of Texas at Austin

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Max Clique and Chromatic Number [FGLSS,…,Hastad]: Max Clique inapproximable to n 1-, any >0, assuming NP ZPP. [LY,…,FK]: Same for Chromatic Number. Can we assume just NP P? Thm: Both inapproximable to n 1-, any >0, assuming NP P. Thm: Derandomized [Khot]: Both inapproximable to n/2 (log n) 1-, some >0, assuming NQ Q. Derandomization tool: disperser.

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Outline Extractors and Dispersers Dispersers and Inapproximability Extractor/Disperser Construction –Additive Number Theory Conclusion

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Weak Random Sources Random element from set A, |A| 2 k. |A| 2 k {0,1} n

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Weak Random Sources Can arise in different ways: –Physical source of randomness. –Condition on some information: Cryptography: bounded storage model. Pseudorandom generators for space-bounded machines. Convex combinations yield more general model, k-source: x Pr[X=x] 2 -k.

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Weak Random Sources Goal: Algorithms that work for any k-source. Should not depend on knowledge of k-source. First attempt: convert weak randomness to good randomness.

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Goal Ext very long weakly random long almost random Should work for all k-sources. Problem: impossible.

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Solution: Extractor [Nisan-Z] Ext very long weakly random long almost random short truly random

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Extractor Parameters [NZ,…, Lu-Reingold-Vadhan-Wigderson] Ext n bits k-source m=.99k bits almost random d=O(log n) truly random Almost random in variation (statistical) distance. Error = arbitrary constant > 0.

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(1- )M K=2 k Graph-Theoretical View of Extractor D=2 d N=2 n M=2 m Think of K=N, M. Goal: D=O(log N). output -uniform x E(x,y 1 ) E(x,y 2 ) E(x,y 3 ) Disperser

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Applications of Extractors PRGs for Space-Bounded Computation [Nisan-Z] PRGs for Random Sampling [Z] Cryptography [Lu, Vadhan, CDHKS, Dodis-Smith] Expander graphs and superconcentrators [Wigderson-Z] Coding theory [Ta-Shma- Z] Hardness of approximation [Z, Umans, Mossel-Umans] Efficient deterministic sorting [Pippenger] Time-space tradeoffs [Sipser] Data structures [Fiat-Naor, Z, BMRV, Ta-Shma]

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Extractor Degree In many applications, left-degree D is relevant quantity: –Random sampling: D=# samples –Extractor codes: D=length of code –Inapproximability of Max Clique: size of graph = large-case-clique-size c D (scaled-down).

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Extractor/Disperser Constructions n=lg N=input length. Previous typical good extractor: D=n O(1). [Ta-Shma-Z-Safra]: –D=O(n log * n), but M=K o(1). –For K=N (1), D=n polylog(n), M=K (1). New Extractor: For K=N (1), D=O(n) and M=K.99. New Disperser: Same, even D=O(n/log s -1 ), s=1- =fraction hit on right side

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Extractor Parameters [Nisan-Z,…, Lu-Reingold-Vadhan-Wigderson] Ext n bits k-source k=lg K.99k bits almost random O(log n) random seed Almost random in variation (statistical) distance. Error = arbitrary constant > 0. [TZS]: For k= (n), lg n + O(log log n) bit seed. New theorem: For k= (n), lg n + O(1) bit seed. (k)

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Dispersers and Inapproximability Max Clique: can amplify success probability of PCP verifier using appropriate disperser [Z]. Chromatic Number: derandomize Feige-Kilian reduction. –[FK]: randomized graph products [BS]. –We use derandomized graph powering. Derandomized graph products of [Alon- Feige-Wigerson-Z] too weak.

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Fractional Chromatic Number Chromatic number (G) N/ (G), (G) = independence number. Fractional chromatic number f (G): (G)/log N f (G) (G), f (G) N/ (G).

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Overview of Feige-Kilian Reduction Poly-time reduction from NP-complete L to Gap-Chromatic Number: –x L f (G) b/c, –x L f (G) > b.

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Overview of Feige-Kilian Reduction Poly-time reduction from NP-complete L to Gap-Chromatic Number: –x L f (G) b/c, –x L (G) b Amplify: G G D, OR product. –(v 1,…,v D ) ~ (w 1,…,w D ) if i, v i ~ w i. – (G D ) = (G) D. – f (G D ) = f (G) D. –Gap c c D. Graph too large: take random subset of V D.

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sM |A| K Disperser picks subset V of V D deterministically D V V x (x 1,x 2,…,x D ) y (y 1,y 2,…,y D ) x x 1 x 2 x 3 y1y1 y y2y2 y3y3 Strong disperser: A i: i (A) sM 1 2 3 2

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sM |A| K (G) < sM (G) < K V V x x i y yiyi If A independent in G, |A| K then ( i) i (A) sM. Since OR graph product.

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Properties of Derandomized Powering If (G) < s|V|, then (G) < K. f (G) f (G D ) = f (G) D.

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Properties of Derandomized Powering If (G) < s|V|, then (G) < K. –If x L, then (G) K N so f (G) N 1- -Since disperser works for any entropy rate >0. f (G) f (G D ) = f (G) D. –If x L, then f (G) N. –Uses D=O((log N)/log s -1 ).

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Extractor/Disperser Outline Condense: Extract:.9 uniform + lg n+O(1) random bits + O(1) random bits

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Extractor for Entropy Rate.9 (extension of [AKS]) G=2 c -regular expander on {0,1} m Weak source input: walk (v 1,v 2,…,v D ) in G –m + (D-1)c bits Random seed: i [D] Output: v i. Proof: Chernoff bounds for random walks [Gil,Kah]

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Condensing with O(1) bits 1.[BKSSW, Raz]: somewhere condenser Some choice of seed condenses. Uses additive number theory [BKT,BIW] 2-bit seed suffices to increase entropy rate. New result: 1-bit seed suffices. Simpler. 2.[Raz]: convert to condenser.

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Condensing via Incidence Graph 1-Bit Somewhere Condenser: –Input: edge –Output: random endpoint Condenses rate to somewhere rate (1+ ), some > 0. –Distribution of (L,P) a somewhere rate (1+ ) source. lines points = F p 2 L P (L,P) an edge iff P on L N 3/2 edges

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Somewhere r-source (X,Y) is an elementary somewhere r-source if either X or Y is an r-source. Somewhere r-source: convex combination of elementary somewhere r-sources.

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Incidence Theorem [BKT] P,L sets of points, lines in F p 2 with |P|, |L| M p 1.9. # incidences I(P,L)=O(M 3/2- ), some >0. lines points = F p 2 L P L P few edges

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Simple Statistical Lemma If distribution X is -far from an r-source, then S, |S|<2 r : Pr[X S]. Proof: take S={x | Pr[X = x] > 2 -r }. 2 -r S FpFp

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Statistical Lemma for Condenser Lemma: If (X,Y) is -far from somewhere r-source, then S supp(X), T supp(Y), |S|,|T| < 2 r, such that Pr[X S and Y T]. Proof: S={s: Pr[X=s] > 2 -r } T={t: Pr[Y=t] > 2 -r }

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Statistical Lemma for Condenser Lemma: If (X,Y) is -far from somewhere r-source, then S supp(X), T supp(Y), |S|,|T| < 2 r, such that Pr[X S and Y T]. Proof of Condenser: Suppose output -far from somewhere r-source. Get sets S and T. I(S,T) 2 r, r = input min-entropy. Contradicts Incidence Theorem.

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Additive Number Theory A=set of integers, A+A=set of pairwise sums. Can have |A+A| < 2|A|, if A=arithmetic progression, e.g. {1,2,…,100}. Similarly can have |A A| < 2|A|. Cant have both simultaneously: –[ES,Elekes]: max(|A+A|,|A A|) |A| 5/4 –False in F p : A=F p

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Additive Number Theory A=set of integers, A+A=set of pairwise sums. Can have |A+A| < 2|A|, if A=arithmetic progression, e.g. {1,2,…,100}. Similarly can have |A A| < 2|A|. Cant have both simultaneously: –[ES,Elekes]: max(|A+A|,|A A|) |A| 5/4 –[Bourgain-Katz-Tao, Konyagin]: similar bound over prime fields F p : |A| 1+, assuming 1 0.

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Independent Sources Corollary: if |A| p.9, then |A A+A| |A| 1+. Can we get statistical version of corollary? –If A,B,C independent k-sources, k.9n, is AB+C close to k-source, k=(1+ )k? (n=log p) –[Z]: under Generalized Paley Graph conjecture. [Barak-Impagliazzo-Wigderson] proved statistical version.

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Simplifying and Slight Strengthening Strengthening: assume (A,C) a 2k-source and B an independent k-source. Use Incidence Theorem. Relevance: lines of form ab+c. How can we get statistical version?

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Simplified Proof of BIW Suppose AB+C -far from k-source. Let S=set of size < 2 k from simple stat lemma. Let points P=supp(B) S. Let lines L=supp((A,C)), where (a,c) line ax+c. I(P,L) |L| |supp(B)| = 2 3k. Contradicts Incidence Theorem.

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Conclusions and Future Directions 1.NP-hard to approximate Max Clique and Chromatic Number to within n 1-, any >0. NQ-hard to within n/2 (log n) 1- some >0. What is the right n 1-o(1) factor? 2.Extractor construction with linear degree for k= (n), m=.99k output bits. Linear degree for general k? 3.1-bit somewhere-condenser. Also simplify/strengthen [BIW,BKSSW,Bo]. Other uses of additive number theory?

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